Can any 3d surface be mapped to a 2d plane? I could see a 3D surface being parameterized by 2 parameters such that the surface is broken up into many infinitesimally thin curves. Is this true even for surfaces that might "fold" over itself or curve back to connect to itself? e.g. a cylinder or a surface like a crumpled up sheet?
 A: How do we define the term surface? By definition, at least for me, a surface in $\mathbb{R}^3$ is a space $M$ for which there exists an injective map $\Phi: D \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$ at each point in $M$. Then there are enough of these patches $\Phi$ for their images to cover all of the surface $M$. So, essentially by definition, a surface is locally in one-one correspondence with the plane. The properties of the maps $\Phi$ can give $M$ more or less structure. For example, the crumpled sheet wouldn't have a smooth patch at crinkle points whereas a cylinder could be given smooth patches. 
Dropping back a dimension is a curve in one-one correspondence with a line? Well, yes. In the plane, there are even curves which fill the space. So, I cannot rule out some subset of a plane from being the image of a curve merely because it does not look like the curves with which I am more familiar. However, if we have some conditions on the curve like regularity etc... then our intuition is restored.
More to the point of your question, a crumpled sheet is a surface (not smooth, if torn not even connected if torn in two) however, the mathematics of such a sheet is challenging. Much more so than a cylinder.
